Change of supersaturation
In gas-solid phase transition, the phase boundary as a function of 
divides the two phases. In equilibrium, if a 
combination falls on the solid side, the matter is in its solid phase. On the other hand, at inequilibrium, for a finite amount of time, a system can be with a combination on the solid side but still remain in gas phase. This is called supersaturation. As shown in the Fig below for the points 
and if they are in gas phase.
The quantitative definition of supersaturation is
,
where 
is the pressure corresponding to the that on the phase boundary line for the temperature 
. This is actually the chemical potential difference between state and state 
in gas phase.
It is useful to know the dependence of on and 
. In other words, 
. Here we will start with the differential form of the relation, i.e. 
and 
. The second partial differential can be found from the definition:
,
while /Tneeds some effort to find.
To find , we look at
![\mu (P,T2)- \mu (P,T1)=[\mu(P,T2)-\mu(P'',T2)]-[\mu(P,T1)-\mu(P',T1)]\\
=[\mu(P,T2)-\mu(P,T1)]-[\mu(P'',T2)-\mu(P',T1)]](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cmu%20%28P%2CT2%29-%20%5Cmu%20%28P%2CT1%29%3D%5B%5Cmu%28P%2CT2%29-%5Cmu%28P%27%27%2CT2%29%5D-%5B%5Cmu%28P%2CT1%29-%5Cmu%28P%27%2CT1%29%5D%5C%5C%0A%3D%5B%5Cmu%28P%2CT2%29-%5Cmu%28P%2CT1%29%5D-%5B%5Cmu%28P%27%27%2CT2%29-%5Cmu%28P%27%2CT1%29%5D)
From Gibbs-Duham equation:
The first term
.
Or
The second term is the chemical potential change along the phase boundary.
Since on the phase boundary, one has
where 
is the enthalpy change between solid and gas phase per mole, which does not vary too much with the temperature.
Again using Gibbs-Duham equation,
.
Therefore,
Or
.
Finally:
.
Or
$d\Delta\mu=RTdlnP-hdlnT$
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